Acta mathematica scientia,Series B ›› 2019, Vol. 39 ›› Issue (3): 764-780.doi: 10.1007/s10473-019-0309-0

• Articles • Previous Articles     Next Articles


Yaozhong HU1, Khoa LÊ2   

  1. 1. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, T6G 2G1, Canada;
    2. Department of Mathematics, South Kensington Campus, Imperial College London, London, SW7 2AZ, United Kingdom
  • Received:2018-03-04 Revised:2019-03-03 Online:2019-06-25 Published:2019-06-27
  • Supported by:
    Y. Hu is supported by an NSERC grant and a startup fund of University of Alberta; K. Lê is supported by Martin Hairer's Leverhulme Trust leadership award.

Abstract: We obtain the Holder continuity and joint Holder continuity in space and time for the random field solution to the parabolic Anderson equation (t-1/2△)u=uW in d-dimensional space, where W is a mean zero Gaussian noise with temporal covariance γ0 and spatial covariance given by a spectral density μ(ξ). We assume that γ0(t) ≤ c|t|-α0 and |μ(ξ)| ≤ c|ξi|-αi or |μ(ξ)| ≤ c|ξ|-α, where αi, i=1,…, d (or α) can take negative value.

Key words: Gaussian process, stochastic heat equation, parabolic Anderson model, multiplicative noise, chaos expansion, hypercontractivity, Hölder continuity, joint Hölder continuity

CLC Number: 

  • 60H15